Note on the Diagonalizable Algebras of PA and ZF

摘要

A proof that the diagonalizable algebras of PA and ZF are not isomorphic ispresented. A number of auxiliary notions are introduced and a number of lemmas are formulated in carrying out the proof. The lemmas are well known and/or easy to believe. The Kalmar elementary functions are chosen as a class of functions of negligibly slow growth elements, to be used as small change, since the proof is to deal with rates of growth of functions. In the given theorem, the PA and ZF pair of theories can be replaced by a wide set of others. The conditions in the two theories under which this replacement is possible are brought together.